An Analytic Solution for 2D Heat Conduction Problems with General Dirichlet Boundary Conditions

Abstract

This paper proposed a closed-form solution for the 2D transient heat conduction in a rectangular cross-section of an infinite bar with the general Dirichlet boundary conditions. The boundary conditions at the four edges of the rectangular region are specified as the general case of space–time dependence. First, the physical system is decomposed into two one-dimensional subsystems, each of which can be solved by combining the proposed shifting function method with the eigenfunction expansion theorem. Therefore, through the superposition of the solutions of the two subsystems, the complete solution in the form of series can be obtained. Two numerical examples are used to investigate the analytic solution of the 2D heat conduction problems with space–time-dependent boundary conditions. The considered space–time-dependent functions are separable in the space–time domain for convenience. The space-dependent function is specified as a sine function and/or a parabolic function, and the time-dependent function is specified as an exponential function and/or a cosine function. In order to verify the correctness of the proposed method, the case of the space-dependent sinusoidal function and time-dependent exponential function is studied, and the consistency between the derived solution and the literature solution is verified. The parameter influence of the time-dependent function of the boundary conditions on the temperature variation is also investigated, and the time-dependent function includes harmonic type and exponential type.